Solved: In 69 and 70, use a CAS as an aid in solving the auxiliary equation. Form the

In Problems 1–14, find the general solution of the given second-order differential equation. \(4 y^{\prime \prime}+y^{\prime}=0)

In Problems 1–14, find the general solution of the given second-order differential equation. \(y^{\prime \prime}-36 y=0\)

In Problems 1–14, find the general solution of the given second-order differential equation. \(y^{\prime \prime}-y^{\prime}-6 y=0\)

In Problems 1–14, find the general solution of the given second-order differential equation. \(y^{\prime \prime}-3 y^{\prime}+2 y=0\)

In Problems 1–14, find the general solution of the given second-order differential equation. \(y^{\prime \prime}+8 y^{\prime}+16 y=0\)

In Problems 1–14, find the general solution of the given second-order differential equation. \(y^{\prime \prime}-10 y^{\prime}+25 y=0\)

In Problems 1–14, find the general solution of the given second-order differential equation. \(12 y^{\prime \prime}-5 y^{\prime}-2 y=0\)

In Problems 1–14, find the general solution of the given second-order differential equation. \(y^{\prime \prime}+4 y^{\prime}-y=0\)

In Problems 1–14, find the general solution of the given second-order differential equation. \(y^{\prime \prime}+9 y=0\)

In Problems 1–14, find the general solution of the given second-order differential equation. \(3 y^{\prime \prime}+y=0\)

In Problems 1–14, find the general solution of the given second-order differential equation. \(y^{\prime \prime}-4 y^{\prime}+5 y=0\)

In Problems 1–14, find the general solution of the given second-order differential equation. \(2 y^{\prime \prime}+2 y^{\prime}+y=0\)

In Problems 1–14, find the general solution of the given second-order differential equation. \(3 y^{\prime \prime}+2 y^{\prime}+y=0\)

In Problems 1–14, find the general solution of the given second-order differential equation. \(2 y^{\prime \prime}-3 y^{\prime}+4 y=0\)

In Problems 15–28, find the general solution of the given higher-order differential equation. \(y^{\prime \prime \prime}-4 y^{\prime \prime}-5 y^{\prime}=0\)

In Problems 15–28, find the general solution of the given higher-order differential equation. \(y^{\prime \prime \prime}-y=0\)

In Problems 15–28, find the general solution of the given higher-order differential equation. \(y^{\prime \prime \prime}-5 y^{\prime \prime}+3 y^{\prime}+9 y=0\)

In Problems 15–28, find the general solution of the given higher-order differential equation. \(y^{\prime \prime \prime}+3 y^{\prime \prime}-4 y^{\prime}-12 y=0\)

In Problems 15–28, find the general solution of the given higher-order differential equation. \(\frac{d^{3} u}{d t^{3}}+\frac{d^{2} u}{d t^{2}}-2 u=0\)

In Problems 15–28, find the general solution of the given higher-order differential equation. \(\frac{d^{3} x}{d t^{3}}-\frac{d^{2} x}{d t^{2}}-4 x=0\)

In Problems 15–28, find the general solution of the given higher-order differential equation. \(y^{\prime \prime \prime}+3 y^{\prime \prime}+3 y^{\prime}+y=0\)

In Problems 15–28, find the general solution of the given higher-order differential equation. \(y^{\prime \prime \prime}-6 y^{\prime \prime}+12 y^{\prime}-8 y=0\)

In Problems 15–28, find the general solution of the given higher-order differential equation. \(y^{(4)}+y^{\prime \prime \prime}+y^{\prime \prime}=0\)

In Problems 15–28, find the general solution of the given higher-order differential equation. \(y^{(4)}-2 y^{\prime \prime}+y=0\)

In Problems 15–28, find the general solution of the given higher-order differential equation. \(16 \frac{d^{4} y}{d x^{4}}+24 \frac{d^{2} y}{d x^{2}}+9 y=0\)

In Problems 15–28, find the general solution of the given higher-order differential equation. \(\frac{d^{4} y}{d x^{4}}-7 \frac{d^{2} y}{d x^{2}}-18 y=0\)

In Problems 15–28, find the general solution of the given higher-order differential equation. \(\frac{d^{5} u}{d r^{5}}+5 \frac{d^{4} u}{d r^{4}}-2 \frac{d^{3} u}{d r^{3}}-10 \frac{d^{2} u}{d r^{2}}+\frac{d u}{d r}+5 u=0\)

In Problems 15–28, find the general solution of the given higher-order differential equation. \(2 \frac{d^{5} x}{d s^{5}}-7 \frac{d^{4} x}{d s^{4}}+12 \frac{d^{3} x}{d s^{3}}+8 \frac{d^{2} x}{d s^{2}}=0\)

In Problems 29–36, solve the given initial-value problem. \(y^{\prime \prime}+16 y=0, \quad y(0)=2, y^{\prime}(0)=-2\)

In Problems 29–36, solve the given initial-value problem. \(\frac{d^{2} y}{d \theta^{2}}+y=0, y(\pi / 3)=0, y^{\prime}(\pi / 3)=2\)

In Problems 29–36, solve the given initial-value problem. \(\frac{d^{2} y}{d t^{2}}-4 \frac{d y}{d t}-5 y=0, y(1)=0, y^{\prime}(1)=2\)

In Problems 29–36, solve the given initial-value problem. \(4 y^{\prime \prime}-4 y^{\prime}-3 y=0, y(0)=1, y^{\prime}(0)=5\)

In Problems 29–36, solve the given initial-value problem. \(y^{\prime \prime}+y^{\prime}+2 y=0, y(0)=y^{\prime}(0)=0\)

In Problems 29–36, solve the given initial-value problem. \(y^{\prime \prime}-2 y^{\prime}+y=0, y(0)=5, y^{\prime}(0)=10\)

In Problems 29–36, solve the given initial-value problem. \(y^{\prime \prime \prime}+12 y^{\prime \prime}+36 y^{\prime}=0, y(0)=0, y^{\prime}(0)=1, y^{\prime \prime}(0)=-7\)

In Problems 29–36, solve the given initial-value problem. \(y^{\prime \prime \prime}+2 y^{\prime \prime}-5 y^{\prime}-6 y=0, y(0)=y^{\prime}(0)=0, y^{\prime \prime}(0)=1\)

In Problems 37-40, solve the given boundary-value problem. \(y^{\prime \prime}-10 y^{\prime}+25 y=0, y(0)=1, y(1)=0\)

In Problems 37-40, solve the given boundary-value problem. \(y^{\prime \prime}+4 y=0, y(0)=0, y(\pi)=0\)

In Problems 37-40, solve the given boundary-value problem. \(y^{\prime \prime}+y=0, y^{\prime}(0)=0, y^{\prime}(\pi / 2)=0\)

In Problems 37-40, solve the given boundary-value problem. \(y^{\prime \prime}-2 y^{\prime}+2 y=0, y(0)=1, y(\pi)=1\)

In Problems 41 and 42, solve the given problem first using the form of the general solution given in (10). Solve again, this time using the form given in (11). \(y^{\prime \prime}-3 y=0, y(0)=1, y^{\prime}(0)=5\)

In Problems 41 and 42, solve the given problem first using the form of the general solution given in (10). Solve again, this time using the form given in (11). \(y^{\prime \prime}-y=0, y(0)=1, y^{\prime}(1)=0)

In Problems 43–48, each figure represents the graph of a particular solution of one of the following differential equations: (a) \(y^{\prime \prime}-3 y^{\prime}-4 y=0\) (b) \(y^{\prime \prime}+4 y=0\) (c) \(y^{\prime \prime}+2 y^{\prime}+y=0\) (d) \(y^{\prime \prime}+y=0\) (e) \(y^{\prime \prime}+2 y^{\prime}+2 y=0\) (f) \(y^{\prime \prime}-3 y^{\prime}+2 y=0\) Match a solution curve with one of the differential equations. Explain your reasoning.

In Problems 43–48, each figure represents the graph of a particular solution of one of the following differential equations: (a) \(y^{\prime \prime}-3 y^{\prime}-4 y=0\) (b) \(y^{\prime \prime}+4 y=0\) (c) \(y^{\prime \prime}+2 y^{\prime}+y=0\) (d) \(y^{\prime \prime}+y=0\) (e) \(y^{\prime \prime}+2 y^{\prime}+2 y=0\) (f) \(y^{\prime \prime}-3 y^{\prime}+2 y=0\) Match a solution curve with one of the differential equations. Explain your reasoning.

In Problems 43–48, each figure represents the graph of a particular solution of one of the following differential equations: (a) \(y^{\prime \prime}-3 y^{\prime}-4 y=0\) (b) \(y^{\prime \prime}+4 y=0\) (c) \(y^{\prime \prime}+2 y^{\prime}+y=0\) (d) \(y^{\prime \prime}+y=0\) (e) \(y^{\prime \prime}+2 y^{\prime}+2 y=0\) (f) \(y^{\prime \prime}-3 y^{\prime}+2 y=0\) Match a solution curve with one of the differential equations. Explain your reasoning.

In Problems 43–48, each figure represents the graph of a particular solution of one of the following differential equations: (a) \(y^{\prime \prime}-3 y^{\prime}-4 y=0\) (b) \(y^{\prime \prime}+4 y=0\) (c) \(y^{\prime \prime}+2 y^{\prime}+y=0\) (d) \(y^{\prime \prime}+y=0\) (e) \(y^{\prime \prime}+2 y^{\prime}+2 y=0\) (f) \(y^{\prime \prime}-3 y^{\prime}+2 y=0\) Match a solution curve with one of the differential equations. Explain your reasoning.

In Problems 43–48, each figure represents the graph of a particular solution of one of the following differential equations: (a) \(y^{\prime \prime}-3 y^{\prime}-4 y=0\) (b) \(y^{\prime \prime}+4 y=0\) (c) \(y^{\prime \prime}+2 y^{\prime}+y=0\) (d) \(y^{\prime \prime}+y=0\) (e) \(y^{\prime \prime}+2 y^{\prime}+2 y=0\) (f) \(y^{\prime \prime}-3 y^{\prime}+2 y=0\) Match a solution curve with one of the differential equations. Explain your reasoning.

In Problems 43–48, each figure represents the graph of a particular solution of one of the following differential equations: (a) \(y^{\prime \prime}-3 y^{\prime}-4 y=0\) (b) \(y^{\prime \prime}+4 y=0\) (c) \(y^{\prime \prime}+2 y^{\prime}+y=0\) (d) \(y^{\prime \prime}+y=0\) (e) \(y^{\prime \prime}+2 y^{\prime}+2 y=0\) (f) \(y^{\prime \prime}-3 y^{\prime}+2 y=0\) Match a solution curve with one of the differential equations. Explain your reasoning.

In Problems 49–58, find a homogeneous linear differential equation with constant coefficients whose general solution is given. \(y=c_{1} e^{x}+c_{2} e^{6 x})

In Problems 49–58, find a homogeneous linear differential equation with constant coefficients whose general solution is given. \(y=c_{1} e^{-5 x}+c_{2} e^{-4 x}\)

In Problems 49–58, find a homogeneous linear differential equation with constant coefficients whose general solution is given. \(y=c_{1}+c_{2} e^{3 x})

In Problems 49–58, find a homogeneous linear differential equation with constant coefficients whose general solution is given. \(y=c_{1} e^{-10 x}+c_{2} x e^{-10 x}\)

In Problems 49–58, find a homogeneous linear differential equation with constant coefficients whose general solution is given. \(y=c_{1} \cos 8 x+c_{2} \sin 8 x\)

In Problems 49–58, find a homogeneous linear differential equation with constant coefficients whose general solution is given. \(y=c_{1} \cosh \frac{1}{2} x+c_{2} \sinh \frac{1}{2} x)

In Problems 49–58, find a homogeneous linear differential equation with constant coefficients whose general solution is given. \(y=c_{1} e^{x} \cos x+c_{2} e^{x} \sin x\)

In Problems 49–58, find a homogeneous linear differential equation with constant coefficients whose general solution is given. \(y=c_{1}+c_{2} e^{-2 x} \cos 5 x+c_{3} e^{-2 x} \sin 5 x)

In Problems 49–58, find a homogeneous linear differential equation with constant coefficients whose general solution is given. \(y=c_{1}+c_{2} x+c_{3} e^{7 x}\)

In Problems 49–58, find a homogeneous linear differential equation with constant coefficients whose general solution is given. \(y=c_{1} \cos x+c_{2} \sin x+c_{3} \cos 3 x+c_{4} \sin 3 x\)

Two roots of a cubic auxiliary equation with real coefficients are \(m_{1}=-\frac{1}{2}\) and \(m_{2}=3+i\). What is the corresponding homogeneous linear differential equation?

Find the general solution of \(y^{\prime \prime \prime}+6 y^{\prime \prime}+y^{\prime}-34 y=0\) if it is known that \(y_{1}=e^{-4 x} \cos x\) is one solution.

To solve \(y^{(4)}+y=0\) we must find the roots of \(m^{4}+1=0\). This is a trivial problem using a CAS, but it can also be done by hand working with complex numbers. Observe that \(m^{4}+1=\left(m^{2}+1\right)^{2}-2 m^{2}\). How does this help? Solve the differential equation.

Consider the boundary-value problem \(y^{\prime \prime}+\lambda y=0, y(0)=0\), \(y(\pi / 2)=0\). Discuss: Is it possible to determine real values of l so that the problem possesses (a) trivial solutions? (b) nontrivial solutions?

In the study of techniques of integration in calculus, certain indefinite integrals of the form \(\int e^{a x} f(x) d x\) could be evaluated by applying integration by parts twice, recovering the original integral on the right-hand side, solving for the original integral, and obtaining a constant multiple k \(\int e^{a x} f(x) d x\) on the left-hand side. Then the value of the integral is found by dividing by k. Discuss: For what kinds of functions f does the described procedure work? Your solution should lead to a differential equation. Carefully analyze this equation and solve for f.

In Problems 65–68, use a computer either as an aid in solving the auxiliary equation or as a means of directly obtaining the general solution of the given differential equation. If you use a CAS to obtain the general solution, simplify the output and, if necessary, write the solution in terms of real functions. \(y^{\prime \prime \prime}-6 y^{\prime \prime}+2 y^{\prime}+y=0\)

In Problems 65–68, use a computer either as an aid in solving the auxiliary equation or as a means of directly obtaining the general solution of the given differential equation. If you use a CAS to obtain the general solution, simplify the output and, if necessary, write the solution in terms of real functions. \(6.11 y^{\prime \prime \prime}+8.59 y^{\prime \prime}+7.93 y^{\prime}+0.778 y=0\)

In Problems 65–68, use a computer either as an aid in solving the auxiliary equation or as a means of directly obtaining the general solution of the given differential equation. If you use a CAS to obtain the general solution, simplify the output and, if necessary, write the solution in terms of real functions. \(3.15 y^{(4)}-5.34 y^{\prime \prime}+6.33 y^{\prime}-2.03 y=0\)

In Problems 65–68, use a computer either as an aid in solving the auxiliary equation or as a means of directly obtaining the general solution of the given differential equation. If you use a CAS to obtain the general solution, simplify the output and, if necessary, write the solution in terms of real functions. \(y^{(4)}+2 y^{\prime \prime}-y^{\prime}+2 y=0\)

In Problems 69 and 70, use a CAS as an aid in solving the auxiliary equation. Form the general solution of the differential equation. Then use a CAS as an aid in solving the system of equations for the coefficients \(c_{i}, i=1,2,3,4\) that result when the initial conditions are applied to the general solution. \(\begin{array}{l}2 y^{(4)}+3 y^{\prime \prime \prime}-16 y^{\prime \prime}+15 y^{\prime}-4 y=0 \\y(0)=-2, y^{\prime}(0)=6, y^{\prime \prime}(0)=3, y^{\prime \prime \prime}(0)=\frac{1}{2}\end{array}\)

In Problems 69 and 70, use a CAS as an aid in solving the auxiliary equation. Form the general solution of the differential equation. Then use a CAS as an aid in solving the system of equations for the coefficients \(c_{i}, i=1,2,3,4\) that result when the initial conditions are applied to the general solution. \(\begin{array}{l}y^{(4)}-3 y^{\prime \prime \prime}+3 y^{\prime \prime}-y^{\prime}=0 \\y(0)=y^{\prime}(0)=0, y^{\prime \prime}(0)=y^{\prime \prime \prime}(0)=1\end{array}\)